The only ones that Don't require a background check (federally) is for private sales, and private citizens do not have access to the NICS background check system.

If you sell a gun to a prohibited person, you are 100% on the hook for and can be charged with a crime. Most people who sell guns in private sales, don't sell to anybody that even looks remotely shady.

Some do, but once again, laws don't and cant stop everything. Requiring mandatory background checks for every sale would not stop a criminal from selling a gun to another criminal.

Also, the private sale is commonly called the "gun show loophole." Purposely called that to trick people into thinking that gun shows don't require background checks. Any FFL holder at of gun show has to run a background check for every firearm sold. A private citizen can sell a gun there without a background check. But he can also meet a buyer in a Walmart parking lot and do the same.

Also, that "loophole", was a compromise both sides made when background checks were first being pushed federally.